A101919 -id:A101919 - cp's OEIS frontend

A101920 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height.

2, 5, 1, 13, 8, 1, 34, 42, 13, 1, 89, 183, 102, 19, 1, 233, 717, 624, 205, 26, 1, 610, 2622, 3275, 1650, 366, 34, 1, 1597, 9134, 15473, 11020, 3716, 602, 43, 1, 4181, 30691, 67684, 64553, 30520, 7483, 932, 53, 1, 10946, 100284, 279106, 342867, 215481

Offset: 1

Emeric Deutsch, Dec 20 2004

A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers, A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields the odd-indexed Fibonacci numbers (A001519).

			T(3,1)=8 because we have HUU'DD, UDUU'DD, UU'DDH, UU'DDUD, UHU'DD, UU'DHD, UU'HDD and UU'UDDD, the up steps starting at odd heights being shown with the prime sign.
Triangle begins:
2;
5,1;
13,8,1;
34,42,13,1;
89,183,102,19,1;

Cf. A006318, A001519, A101919.

G := 1/2/(-t*z+t*z^2)*(-1+3*z-t*z-z^2+sqrt(1-6*z-2*t*z+11*z^2+2*t*z^2-6*z^3+t^2*z^2-2*t*z^3+z^4)): Gser:=simplify(series(G,z=0,13)):for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields the sequence in triangular form

G.f.=G=G(t, z) satisfies tz(1-z)G^2-(1-3z+tz+z^2)G+1-z=0.

cp's OEIS Frontend

A101920 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height.

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